32 Prof. A. Gray on Canonical 



[The other relations (6) and (7) of § 6 ma)' be proved in 

 the same manner.] 



11. Starting from the relations 



which follow at once from 



^ = -H (2) 



and using the variable t instead of a i in equations corre- 

 sponding to (2), (4), (5) of § 10, Ave pass to the canonical 

 equations 



clt ~ -dq j dt~"dp' ' ' ' ' [ } 



in which the meanings of dH/d</, dH/^yj have been changed, 

 as explained above. 



12. In the light of the Hamiltonian function IS we now 

 consider the variation of the trajectory of a holonomous 

 system. Since, § 4, Sj( = S — S ) involves only the initial 

 and final coordinates, with t and r, it has the same value for 

 any passage of the system from a given initial to a given 

 final configuration without variation of the time of starting 

 or the time of arrival. If we consider a possible succession 

 of configurations we can derive another succession from it 

 by the process of variation ; but, as is now well understood, 

 the derived succession may or may not, according to the 

 condition imposed on the system, be itself a possible one. 



Let the configuration at time t be defined by the co- 

 ordinates qi, q 2 , ... ., q k . Then it is possible to assign 

 another set of coordinates, qi + Bq l9 qi + Sq a , . . . ., q k + ^q k ^ 

 for the same instant t, and if the system be holonomous, that 

 is if qi, q 2 , . . . ., q k be the coordinates left independent after 

 satisfaction of the (finite) kinematical relations, 8q l9 Sq 2 , . . ., 

 8q k are any perfectly arbitrary small changes in the co- 

 ordinates. In this case the succession of configurations 

 given by the different values of t will be a possible one for 

 the system, if Sq^ 8q 2 , . . . ., 8q k are continuous functions 

 of t, such that at every point of the locus of the sequence of 

 positions (taken by the representative point P whose position 

 in &- dimensional space is defined by q x , q 2 , . . . ., q k ) there is 

 a definite derivative. 



The system is still holonomous if the kinematical con- 

 ditions involve t explicitly, provided these conditions are 



